MATH 267 Midterm: MATH 267 2012 Winter Test 1
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[12 points] let f (t) be a periodic function with f (t) = |t| for 2 < t 2 and period t = 4. (a) [3 points] sketch the graph of f (t). Is f (t) odd, even or neither? (b) [6 points] find the real fourier series of f (t). (c) [3 points] use part (b) to calculate the in nite sum. [12 points] find the solution u(x, t) of the wave equation: utt = 4uxx, u(0, t) = u(3, t) = 0, u(x, 0) = 5 sin(cid:0) 4 ut(x, 0) = 1, t > 0. [15 points] (a) [5 points] find the fourier transform of (b) [5 points] find the inverse fourier transform of f (t) = 9 + (t + 1)2 (c) [5 points] find the inverse fourier transform of. 2 + i (1 + i ) (3 + i ) bg( ) = ei4 bh( ) = cos(cid:16) .
